A piecewise Korn inequality in SBD and applications to embedding and density results. (English) Zbl 1391.74227
Summary: We present a piecewise Korn inequality for generalized special functions of bounded deformation (\(GSBD^2\)) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that \(GSBD^2\) functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in \(GSBD^2\) in dimension two. Moreover, for all \(d \geq 2\) we show \(GSBD^2(\Omega) \subset (GBV(\Omega;{\mathbb R}))^d\) and the embedding \(SBD^2(\Omega) \cap L^\infty(\Omega;{\mathbb R}^d) \hookrightarrow SBV(\Omega;{\mathbb R}^d)\) into the space of special functions of bounded variation (\(SBV\)). Finally, we present a Korn-Poincaré inequality
for functions with small jump sets in arbitrary space dimension.
MSC:
| 74R10 | Brittle fracture |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 70G75 | Variational methods for problems in mechanics |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
Keywords:
functions of bounded deformation; Korn’s inequality; Korn-Poincaré inequality; coarea formula; brittle materials; variational fractureReferences:
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